Stabilization of discrete time stochastic system with input delay and control dependent noise

In this paper, we introduce a delay dependent Lyapunov equation (DDLE) approach to study the mean square stabilization for discrete time stochastic system with both input delay and control dependent noise. The innovative contributions of this paper are twofold. First, for a general stochastic system with input delay and multiplicative noises, we derive a necessary stabilizing condition based on a coupled Lyapunov equation (CLE). Second, we present a set of necessary and sufficient stabilizing conditions for the considered stochastic system. We show that the stochastic system is stabilizable is equivalent to that the DDLE has a positive definite solution. In this case, the constructed CLE is equivalent to the DDLE. Moreover, based on the Lyapunov stabilizing result, we further derive a spectrum stabilizing criterion. To confirm the effectiveness of our theoretic results, two illustrative examples are included.